Title: Solution Manual for Elasticity: Theory, Applications, and Numerics – 5th Edition by Martin H. Sadd | Complete Step-by-Step Solutions for Stress and Strain Analysis, Continuum Mechanics, Elastic Deformation, Energy Methods, Boundary Value Problems,

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ALL 16 CHAPTERS COVERE




SOLUTIONS MANUAL

,Table of contents
Part 1: Foundations and elementary applications

1. Mathematical Preliminaries

2. Deformation: Displacements and Strains

3. Stress and Equilibrium

4. Material Behavior – Linear Elastic Solids

5. Formulation and Solution Strategies

6. Strain Energy and Related Principles

7. Two-Dimensional Formulation

8. Two-Dimensional Problem Solution

9. Extension, Torsion, and Flexure of Elastic Cylinders

Part 2: Advanced applications

10. Complex Variable Methods

11. . Anisotropic Elasticity

12. Thermoelasticity

13. . Displacement Potentials and Stress Functions: Applications to Three-Dimensional Problems

14. Nonhomogeneous Elasticity

15. . Micromechanics Applications

16. Numerical Finite and Boundary Element Methods

,1


1-1.

(a) aii a11 a22 a33 1 4 1 6 (scalar)
aij aij a11a11 a12 a12 a13 a13 a 21a 21 a22 a22 a23 a23 a31a31 a32 a32 a33 a33
1 1 1 0 16 4 0 1 1 25 (scalar)
⎡1 1 1⎤⎡ 1 1 1⎤ ⎡1 6 4 ⎤
⎢ ⎥⎢ ⎥ ⎢ ⎥
aij a jk 4 4 18 10⎥ (matrix)
⎢0 2⎥⎢ 2⎥
0 ⎢0
⎢⎣0 1 1⎥⎦ ⎢⎣0 1 1⎥ ⎢⎣0 5 3 ⎥⎦
⎦
⎡3⎤
⎢ ⎥
aij b j ai1b1 a i 2b 2 ai3b3 ⎢4⎥ (vector)
⎢⎣2⎥⎦
aij bib j a11b1b1 a12b1b2 a13b1b3 a21b2b1 a 22b2b2 a23b2b3 a31b3b1 a32b3b2 a33b3b3
1 0 2 0 0 0 0 0 4 7 (scalar)
⎡b1b1 b1b2 b1b3 ⎤ ⎡1 0 2⎤
⎢ ⎥ ⎢ ⎥
bi b j b2b2 b2 b3
⎢b2 0 0⎥ (matrix)
b1 ⎥ ⎢0
⎢⎣b3 b1 b3b2 b3b3 ⎢⎣2 0 4⎥⎦
⎥⎦
bibi b1b1 b 2b 2 b3b3 1 0 4 5 (scalar)

(b) aii a11 a22 a33 1 2 2 5 (scalar)
aij aij a11a11 a12a12 a13a13 a21a21 a22 a22 a23a23 a31a31 a32 a32 a33a33
1 4 0 0 4 1 0 16 4 30 (scalar)
⎡1 2 0⎤⎡ 2 0⎤ ⎡1 6 2⎤
⎢ 1 ⎥ ⎢ ⎥
⎥⎢
aij a jk 2 2 1⎥ ⎢0 4⎥ (matrix)
⎢0 1⎥⎢ 8
0
⎢⎣0 4 4 2⎥⎦ ⎢⎣0 8⎥⎦
2⎥⎦ ⎢ ⎡4⎤ 16
⎣0 ⎢ ⎥
a ijb j ai1b1 ai 2b2 ai3b3 ⎢3⎥ (vector)
⎢⎣ 6⎥⎦
aijbib j a11b1b1 a12b1b2 a13b1b3 a21b2b1 a22b2b2 a23b2b3 a31b3b1 a32b3b2 a33b3b3
4 4 0 0 2 1 0 4 2 17 (scalar)
⎢
⎡b1b1

,b1b2 b1b3 ⎡4
⎤
⎥ 2

2⎤
⎢ ⎥
bi bj b2b2 b2b3 1⎥ (matrix)
⎥
⎢b2b1 ⎢2 1
⎢⎣b3 b1 b3b2 b3b3 ⎢⎣2 1 1⎥⎦
⎥⎦
bibi b1b1 b2b2 b3b3 4 1 1 6 (scalar)




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,3


(c) aii a11 a22 a33 1 0 4 5 (scalar)
aij aij a11a11 a12 a12 a13a13 a21a21 a22 a22 a23a23 a31a31 a32 a32 a33a33
1 1 1 1 0 4 0 1 16 25 (scalar)
⎡1 1 1⎤⎡ 1 1 ⎡2 2 7 ⎤
⎢ 1⎤ ⎢ ⎥
⎥⎢
⎥
aij a jk ⎢1 2⎥⎢ 1 2⎥ ⎢1 9 ⎥ (matrix)
0 3
0
⎢⎣0 1 4⎥⎦ ⎢⎣0 1 4⎥⎦ ⎢⎣ 4 18⎥⎦
⎡2 1
⎤
⎢ ⎥
a ijb j ai1b1 ai 2b2 ai3b3 ⎢1⎥ (vector)
⎢⎣1⎥⎦
aijbib j a11b1b1 a12b1b2 a13b1b3 a21b2b1 a22b2b2 a23b2b3 a31b3b1 a32b3b2 a33b3b3
1 1 0 1 0 0 0 0 0 3 (scalar)
⎡b1b1 b1b b1b3 ⎤ ⎡1 1 0⎤
⎢ 2 ⎥ ⎢ ⎥
bi bj b2b2 b2b3 0⎥ (matrix)
⎥
⎢b 2b 1 ⎢ 1 1
⎢⎣b3 b1 b3b2 b3b3 ⎢⎣ 0 0⎥⎦
⎥⎦ 0
bibi b1b1 b2b2 b3b3 1 1 0 2 (scalar)



1-2.
1 1
(a) aij  a ji )  a ji )
(a (a
2 ij 2 ij
⎡ 2 1 ⎡ 0 1 1⎤
1⎤
1⎢ 1⎢
1 3⎥ 1 0 1⎥

8
2⎢ ⎥ 2⎢ ⎥
⎢⎣1 3 2⎥⎦ ⎢⎣ 1 0⎥⎦
1
clearlya(ij ) and a[ij ] satisfy the appropriate conditions

1 1
(b) aij  a ji )  a ji )
(a (a
ij ij
2 2

, ⎡2 2 ⎡ 02 0 ⎤
0⎤
1⎢ 1
2 4 5⎥ 3⎥
⎢ 2 0
2 ⎢ ⎥ 2 ⎢ ⎥
⎢⎣0 5 4⎥⎦ ⎢⎣ 3 0 ⎥⎦
0
clearlya(ij ) and a[ij ] satisfy the appropriate conditions




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,5


1 1
(c) aij  a ji )  a ji )
(a (a
ij
2 2 ij
⎡2 2 ⎡ 0 0 1⎤
1⎢⎤
1
2 0 0 1⎥
⎥
1 ⎢3
0
2 ⎢ ⎥ 2⎢ ⎥
⎢⎣1 3 8⎥⎦ ⎢⎣ 1 0⎥⎦
1
clearlya(ij ) and a[ij ] satisfy the appropriate conditions


1-3.
aijbij a jib ji aijbij
2aijbij 0 aijbij 0
⎛ ⎞ T

1 ⎤⎡ 0 1 1 ⎤
⎜ ⎡2 ⎟⎟
From Exercise1- 2(a) : a a 1 3⎥⎢ 1 1⎥ 0
1 tr⎜ ⎢18 0
(ij ) [ij ] 4 ⎜ ⎥⎢ ⎥
⎢ 1 0⎥ ⎟⎟
3  ⎠
⎜

⎢1 2⎥⎦ ⎢⎣ 2 0 ⎤T ⎞
⎝ 
1
⎛
 2 0 ⎤⎡
⎡2
0
1
From Exercise1- 2(b) : a a 4 ⎟ 0
tr ⎜ ⎢2 5⎥⎢ 3⎥
2 0
(ij ) [ij ] 4 ⎜⎢ ⎥⎢ ⎥ ⎟
⎜ ⎢0 5 3 ⎥
0  ⎟
⎝ 4⎥⎦ ⎢⎣ ⎠
⎛ 0 T
⎞
⎜ ⎡2 2 1 ⎤⎡ 0 ⎟
From Exercise1- 2(c) : a a 1 ⎜ ⎢2 0
tr 1 ⎟ 0
0 3⎥⎢ ⎤
0 0
1
⎥
(ij ) [ij ] 4 ⎜ ⎥⎢ ⎥
⎢ 1 0⎥ ⎟⎟
⎜
3  ⎠

⎢1 8⎥⎦ ⎢⎣
⎝
1



1-4. 13 a3 ⎤
⎡ a
11 1 12 a2

, ⎡a1 ⎤
⎢ ⎥ ⎢ ⎥
ij a j i1a1 i 2 a2 i3 a3 a
⎢ 21 1 22 a2 23 a3 ⎥ ⎢a2 ⎥ ai
⎢⎣ 31a1 32 a 2
⎥
33 a 3 ⎦
⎢
⎣ a3 ⎥⎦
⎡ a
11 11 12 a21 a
11 12 12 a 22 11a13 12 a23 a ⎤
13 33
⎥
13a31 a
⎢ 13 32

ij a jk ⎢ ⎥
⎢⎣ ⎥⎦
⎡a11 a12
⎢
 a21 a22
⎢
⎢⎣a31 a32
a13 ⎤
⎥
a 23 a ij
⎥
a33 ⎥⎦




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,7



1-5.
det(aij ) ijk a1ia2 ja3k a a a
123 11 22 33 a a a
231 12 23 31 a a a
312 13 21 32

a a a
321 13 22 31 132a11a 23a32 a a a
213 12 21 33

a11a22a33 a12a23a31 a13a21a32 a13a22a31 a11a23a32 a12a21a33
a11 (a22a33 a23a32 ) a12 (a21a33 a23a31 ) a13 (a21a32 a22a31 )
a11 a12 a13
a21 a22 a23
a31 a32 a33


1-6.
⎡1 0 0 ⎤
45o rotation about x 1 - axis Qij 2 /2 ⎥

⎢0 2 /2 ⎥⎦

⎢⎣
0
⎡1 0 0 ⎡ 1 ⎤
⎤⎡1
⎤
From Exercise1-1(a) : 2 /2 ⎥⎢0⎥ ⎢ 2⎥
b Qb ⎢0
i ij j ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢⎣0 2/2 2/ ⎣⎢ 2 ⎥⎦
2⎥⎦ ⎢⎣2⎥⎦
⎡1 0 0 ⎤⎡1 1 1⎤⎡1 0 ⎤T ⎡1 2 0⎤
⎢ ⎥⎢ 0 ⎥ ⎢ ⎥
⎥⎢
ai Qip Q jq a pq 2/2 2 ⎥⎢0 2/2
j ⎢0 ⎥⎢0 2/ 4 1⎥
⎢⎣0 2 /2 2 /2 ⎢0 ⎥
2 1 
1⎥⎦ ⎢⎣0 2⎥
⎢0
4 0 ⎤⎡2⎤ 2/ ⎣
⎥⎦ ⎢⎣0 1 2⎥⎦
⎡1 0 ⎡ 2 ⎤
From Exercise1-1(b) : 2/2 ⎥⎢ 1⎥ ⎢ 2⎥
b Qb ⎢0
i ij j ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢⎣0 2 /2 2/ ⎢⎣ 0 ⎥⎦
2⎥⎦ ⎢⎣1⎥⎦
⎡1 0 0 2 0⎤⎡1 0 ⎤ 2 2⎤
⎢ ⎡T1 ⎥
⎤⎡ 0
1 ⎥⎢ ⎥ ⎢
⎥⎢
ai Qip Q jq a pq 2/2 14⎥⎢0 2/2 2 /2
j ⎢0 ⎥⎢0 2/ ⎣0 0
⎢⎣0 2 /2 2 2⎥⎦ ⎢ 2⎥⎦ ⎢⎣0
⎡1 0 ⎤⎡1⎤ ⎡

, ⎥ ⎢0 4.5 1.
⎢ 1.5 5⎥ 0
⎥⎦ ⎣
0
1 ⎤ 
.5⎥
From Exercise1-1(c) : 2 /2 2 /⎢2⎥⎢1⎥ 2 /2 ⎥
b Qb ⎢0
i ij j ⎢ ⎥⎢ ⎥ ⎢ ⎥
⎢⎣0 2/2 2/ ⎢⎣ ⎥⎦
2⎥⎦ ⎢⎣0⎥⎦
⎡1 0 0 ⎤⎡ 1 1 1⎤⎡ 1 0 0 ⎤
T
 1 2 0 
⎢ ⎥⎢ ⎥⎢ ⎥  
ai j Qip Q jq a pq ⎢0
2/2 ⎥⎢1 2 ⎥⎢ 2/2 2/  3.5 2.5
2 /2 0 2 /2 1.5  ⎥
0.5
⎢⎣0 0 2⎥ 
⎢
⎥⎦ ⎢⎣0 1 4⎥⎦ ⎢ ⎥⎦ 
⎣0




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